Python: How can I change custom property for multiple objects at once?
Instead of accessing the ECollider like this: object.ECollider now I access it this way: object['ECollider'] and everything works fine. Full code modification:

def set_collider(self, context):
    selected_objs = context.selected_objects
    collider = context.active_object['ECollider']
    for obj in selected_objs:
        if obj != context.active_object:
            obj['ECollider'] = collider

bpy.types.Object.ECollider = EnumProperty(
    items=[('NC', 'None', ''), ('BC', 'Box Collider', ''), ('SC', 'Sphere Collider', '' ), ('CC', 'Capsule Collider', '')], name="Collider", update=set_collider
)